Why is gravitational potential energy negative for the equation of Ep equal -GMm over r?

Answer 1

Imagine a gravity well. The farther you move away from the center, the "flatter" space becomes. The "gravitational potential" to which your question refers is zero at infinty. If the universe consisted of only the earth and a marble an infinite distance away, its PE would be zero. If it were moving toward the earth, its KE would increase the closer it became, until it struck the planet. Energy conservation requires that, as the kinetic form increases, the potential form decreases. If Ep is decreasing from zero, then it must be negative. Be aware that this is a matter of convention--like electrons being negative and number lines like the "x-axis" increasing to the right. What's important is consistency within the system. In the case of potential energy, it is generally the change or difference in energy that matters, not an isolated value at an arbitrary point. Other conventions can work, but this has become the consensus: "-"GMm/r becomes greater as the masses move (via work) away from each other. This fits with the model, as potential energy is increasing towards zero. Confusion often comes from near the surface of earth problems. There we use U = mgy, and we use ground level as zero PE (your Ep) and increase potential energy to the plus side of zero as we move away from the center of the earth's gravity well. This works, so long as you do not move in the y-direction (orthodox: y increases as you go up) enough to cause the value of 'g' to vary--which is what happens over astronomical distances. When that happens, you need to switch BOTH formulas AND concepts of where Ep is zero, but NOT of in which direction it is increasing. Ep ALWAYS grows greater as you do work against gravity; i.e., move away from the center of the gravity well (often earth). Since students are usually first taught U = mgy, it becomes harder to grasp the "zero value" for potential energy being at the other end--away from the gravity well's center. This forces us to consider it to be growing ever more negative (getting smaller) as it nears the gravitational point source. It's convention because it works, and I hope it makes sense if it is explained as such. Final note, it would not matter if we were to add any constant to the potential, so we can actually set the potential at infinity to be any value (as opposed to the usual case of setting it to zero), that would make the potential U = -GMm/r + C, which is more complicated. So the simplest case is just to set the constant equal to zero. So why can't we set the potential at the origin (the bottom of the well) to zero instead? That would be nice since, if we set the bottom of the well to zero, then the potential would be positive which would "look" better. Most of the potentials discussed in elementary mechanics are positive, except for long ranged ones like gravity, which most likely is the reason why people don't appreciate gravity's potential very well. This is because the bottom of the well is infinitely deep and adding any finite constant can't lift it to zero.